NONUNIFORM TORSION, UNIFORM SHEAR AND TIMOSHENKO ...

(1)ΕΕ..ΜΜ..ΠΠ.. Σχολή Πολιτικών Μηχανικών Εφαρμοσμένη Ανάλυση Ραβδωτών και Επιφανειακών Φορέων

NONUNIFORM TORSION, NONUNIFORM TORSION, UNIFORM SHEAR AND UNIFORM SHEAR AND

TIMOSHENKO THEORY OF TIMOSHENKO THEORY OF ELASTIC HOMOGENEOUS ELASTIC HOMOGENEOUS

ISOTROPIC PRISMATIC BARSISOTROPIC PRISMATIC BARS

COURSE : APPLIED STRUCTURAL ANALYSIS OF FRAMED AND SHELL STRUCTURES (A1)

Lecturer :Ε. J. SapountzakisProfessor NTUA

NATIONAL TECHNICAL UNIVERSITY OF ATHENSNATIONAL TECHNICAL UNIVERSITY OF ATHENSSCHOOL OF CIVIL ENGINEERING

INTER-DEPARTMENTAL POSTGRADUATE COURSES PROGRAMMES«ΔΟΜΟΣΤΑΤΙΚΟΣ ΣΧΕΔΙΑΣΜΟΣ ΚΑΙ ΑΝΑΛΥΣΗ ΚΑΤΑΣΚΕΥΩΝ»

“ANALYSIS AND DESIGN OF EARTHQUAKE RESISTANT STRUCTURES”


Warping due to transverse shear

Warping due to torsion

Eccentric Transverse Eccentric Transverse Shear Loading Shear Loading QQ

Shear LoadingShear Loading QQ Direct Torsional LoadingDirect Torsional Loading ΜΜt==QQ••ee)

Nonuniform shear stress flow due to

Q

Nonuniform shear stress flow

due to Mt

((DirectDirect: : Equilibrium Torsion,Equilibrium Torsion, IndirectIndirect: : Compatibility TorsionCompatibility Torsion))


WARPING OF A HOLLOW SQUARE CROSS SECTIONWARPING OF A HOLLOW SQUARE CROSS SECTION

Due to Torsion

Due to Shear


CROSS SECTIONS EXHIBITING SMALL AND SIGNIFICANT WARPING

SMALL WARPINGSMALL WARPING(Closed shaped cross sections)

INTENSE WARPINGINTENSE WARPING(Open shaped cross sections)


COMPARISON OF TORSIONAL DEFORMATIONS OF THIN WALLED TUBES COMPARISON OF TORSIONAL DEFORMATIONS OF THIN WALLED TUBES HAVING CLOSED AND OPEN SHAPED CROSS SECTIONSHAVING CLOSED AND OPEN SHAPED CROSS SECTIONS

θ0

θ1

rm

t

Distribution of primary shear stresses

w θ1

θ0

rm

t

Distribution of primary shear stresses

tο

100

00

1

300Close Open

m

t t

r mmt m

I

m

I

=

=

=


CLASSIFICATION OF CLASSIFICATION OF TORSION AS A STRESS STATETORSION AS A STRESS STATE

Direct Torsion(Equilibrium Torsion)

Indirect Torsion(Compatibility Torsion)

Cracking due to creep and shrinkage effects Significant reduction of torsional rigidity

Bridge deck of box shaped cross section curved in plan (Permanent) torsional

loading due to self-weight


Uniform ShearUniform Shear –– TorsionTorsion

Shear Stresses Exclusively(Saint–Venant, 1855)

Classification of shear & torsion according to longitudinal variClassification of shear & torsion according to longitudinal variation of warping ation of warping (UNIFORM (UNIFORM -- NONUNIFORM SHEAR AND TORSION)NONUNIFORM SHEAR AND TORSION)

• Transverse Load, Twisting Moment: Constant

• Warping (Q, Mt): Free (Not Restrained)

Nonuniform ShearNonuniform Shear –– Torsion Torsion (Wagner, 1929)

Shear Stresses (Primary (St. Venant))

Stresses due to Warping (Normal stresses& Secondary shear stresses)

• Transverse Load, Twisting Moment: Variable

• Warping (Q, Mt): Restrained


Mt

l

b

h

E, G

Primary Shear Stresses (torsional loading)

Secondary (Warping) Shear stresses (torsional loading)

Closed Bredt stress flow, 1896Complex distribution in thick walled cross

sections (thin-walled: Vlasov, 1963)

SHEAR STRESS DISTRIBUTION (NONUNIFORM TORSION)SHEAR STRESS DISTRIBUTION (NONUNIFORM TORSION)


PRANDTLPRANDTL’’S MEMBRANE ANALOGYS MEMBRANE ANALOGY

Saint Venant (uniform) torsion has been Saint Venant (uniform) torsion has been ““depicteddepicted”” byby PrandtlPrandtl ((19031903)) through the through the membrane analogy: Uniform torsion and membrane problems are descmembrane analogy: Uniform torsion and membrane problems are described from ribed from analogous boundary value problemsanalogous boundary value problems. .

Rectangular Cross section

φ1 max φ3

φ3

φ2

Membrane φ3=0

(φ1>φ2) T-shaped

cross section

Membrane

φ1 max

The deformed membrane offers the following informationThe deformed membrane offers the following information: :

• Contours correspond to the directions of the trajectories of sheContours correspond to the directions of the trajectories of shear stressesar stresses

• The slopes of the deformed membrane correspond to the values of The slopes of the deformed membrane correspond to the values of shear stressesshear stresses

• The volume of the deformed membrane corresponds to St. The volume of the deformed membrane corresponds to St. VenantVenant’’ss torsion constanttorsion constant


• Stress State&

• Strain State: Nonuniform TorsionNonuniform Torsion

Seven degrees of freedom (14x14 [K])• Additional dof.: Twisting curvature• Additional stress resultant: Warping

moment

Warping due to torsion

WarpingWarping duedue toto shearshear <<<< WarpingWarping duedue toto torsiontorsion

Warping due to shear

SHEAR FORCESHEAR FORCE TWISTING MOMENTTWISTING MOMENT

(Significant in Open Shaped Cross Sections)

• Stress State (Stress field):Uniform ShearUniform Shear

• Strain/Deformation State:

Shear Deformation CoefficientsShear Deformation Coefficients Indirect account of warping deformation (Timoshenko,

1922)


Problem description of Nonuniform Torsion & Uniform ShearProblem description of Nonuniform Torsion & Uniform Shear

Thin Walled Beam theory(Vlasov theory, 1964)

Generalized Beam Theory(Schardt, 1966)

Technical Beam Theory•• Limited set of cross sections (of simple geometry)Limited set of cross sections (of simple geometry)•• Warping restraints are ignoredWarping restraints are ignored•• Compatibility equations are not employedCompatibility equations are not employed•• Stress computations are performed studying equilibrium Stress computations are performed studying equilibrium

of a finite segment of a bar and not equilibrium of an of a finite segment of a bar and not equilibrium of an infinitesimal material pointinfinitesimal material point (3d elasticity)(3d elasticity)

•• Valid for thin walled cross sections (Midline Valid for thin walled cross sections (Midline employed)employed)

•• Warping restraints are taken into accountWarping restraints are taken into account•• ReliabilityReliability: : Depends on thickness of shell elements Depends on thickness of shell elements

comprising the beamcomprising the beam

•• Valid for arbitrarily shaped cross sectionsValid for arbitrarily shaped cross sections(Thick or Thin walled)

•• Warping restraints are taken into accountWarping restraints are taken into account•• BVPsBVPs formulated employing theory of formulated employing theory of 33D elasticityD elasticity•• Numerical solution of Numerical solution of BVPsBVPs


Analysis of Bars and Bar Assemblages Direct Stiffness Method

• Application of 12x12 Stiffness Matrix (6 dofs per node)

• Approximate Computation of Torsion Constant

• Approximate Computation of Shear Deformation Coefficients

• Approximate Computation of stresses due to shear and torsion

Everyday Engineering PracticeEveryday Engineering Practice:

InaccuraciesInaccuracies Non conservative Design (sometimes)Non conservative Design (sometimes)

Plane Steel FramePlane Steel

GridSpatial Steel

Frame


ASSUMPTIONS OF ELASTIC THEORY OF TORSION•• The bar is straightThe bar is straight..•• The bar is prismaticThe bar is prismatic..•• The barThe bar’’s longitudinal axis is subjected to twisting exclusivelys longitudinal axis is subjected to twisting exclusively..•• Distortional deformations of the cross section are not allowed (Distortional deformations of the cross section are not allowed (cross sectional shape is not cross sectional shape is not altered during deformationaltered during deformation ((γγ2323=0=0, , distortiondistortion neglectedneglected)). . •• Twisting rotation is considered small: Circular arc displacementTwisting rotation is considered small: Circular arc displacements are approximated with s are approximated with the corresponding displacements along the chordsthe corresponding displacements along the chords..•• The material of the bar is homogeneous, isotropic, continuous (nThe material of the bar is homogeneous, isotropic, continuous (no cracking) and linearly o cracking) and linearly elastic: Constitutive relations of linear elasticity are validelastic: Constitutive relations of linear elasticity are valid..•• The distribution of stresses at the bar ends is such so that allThe distribution of stresses at the bar ends is such so that all the aforementioned the aforementioned assumptions are validassumptions are valid. .

Especially forEspecially for (unrestrained) uniform(unrestrained) uniform (Saint Venant) torsion(Saint Venant) torsion the following assumption is the following assumption is also validalso valid::•• Longitudinal displacements (warping) are not restrained and do nLongitudinal displacements (warping) are not restrained and do not depend on the ot depend on the longitudinal coordinate (every cross section exhibits the same wlongitudinal coordinate (every cross section exhibits the same warping deformations)arping deformations)..


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSION

Displacement FieldDisplacement Field

2x3x

L

Ε, G

b=L

a=0

( )1t tm m x=

1x

2x

3x

Ο •

2 0t =

3 0t =

ΓΚ+1

(Ω)

ΓΚ

Γ1 Γ2

∪ 1

1

+

=Γ=Γ

K

j j

Ρ

Ρ′

ω

ω 3u

2u1θ(x )

n

t

s

M

3x

2Mx

3Mx

2x

M: C.of T.=S.C. (ν=0)

2 1 2 3 1 1 3 1 1u ( x ,x ,x ) (PP )sin (MP ) (x )sin x (x )ω θ ω θ′= − = − = −

3 1 2 3 1 1 2 1 1u ( x ,x ,x ) (PP )cos (MP ) (x )cos x (x )ω θ ω θ′= = =

( ) ( ) ( )1 1 2 3 1 1 M 2 3u x ,x ,x x x ,xθ ϕ′= + ⋅



Components of the Infinitesimal Strain TensorComponents of the Infinitesimal Strain Tensor

( ) ( )111 1 1 M 2 3

1

ux x ,x

xε θ ϕ

∂ ′′= = ⋅∂

222

2

u0

xε

∂= =∂

333

3

u0

xε

∂= =∂

( )1 212 1 1 3

2 1 2

u ux x

x x xΜϕγ θ

⎛ ⎞∂ ∂ ∂′= + = ⋅ −⎜ ⎟∂ ∂ ∂⎝ ⎠

3223

3 2

uu0

x xγ

∂∂= + =∂ ∂

( )3113 1 1 2

3 1 3

uux x

x x xΜϕγ θ

⎛ ⎞∂∂ ∂′= + = +⎜ ⎟∂ ∂ ∂⎝ ⎠

11 0 St .V .ε = →



Components of the Cauchy Stress TensorComponents of the Cauchy Stress Tensor ((νν=0)=0)

( )( ) ( ) ( ) ( ) ( )11 11 22 3 13 1 M 2 3E 1

1E x x ,x

1 2τ ν ε ν ε ϕε

ν νθ⎡ ⎤= − + + ′′ ⋅=⎣ ⎦−⋅

+

( )( ) ( ) ( )22 22 11 33E 1 0

1 1 2τ ν ε ν ε ε

ν ν⎡ ⎤= − + + =⎣ ⎦+ −

( )( ) ( ) ( )33 33 11 22E 1 v v 0

1 v 1 2vτ ε ε ε⎡ ⎤= − + + =⎣ ⎦+ −

( ) M1 1 3

212 12 G x x

xG ϕτ γ θ

⎛ ⎞∂′⋅ ⋅= ⋅ −⎜ ⎟∂⎝ ⎠=

32 32G 0τ γ= ⋅ =

( ) M1 1 2

331 31 G x x

xG ϕτ γ θ

⎛ ⎞∂′⋅ ⋅= ⋅ +⎜ ⎟∂⎝ ⎠=

11 0 St.V .τ = →



Differential Equilibrium Equations ofDifferential Equilibrium Equations of 33DD ElasticityElasticity(Body forces neglected)

( )

( )

M1 1 3

2

M1 1 2

3

G x x 0x

G x x 0x

ϕθ

ϕθ

⎫⎛ ⎞∂′′⋅ ⋅ − = ⎪⎜ ⎟∂⎝ ⎠ ⎪ →⎬⎛ ⎞∂ ⎪′′⋅ ⋅ + =⎜ ⎟ ⎪∂⎝ ⎠ ⎭

Not Satisfied!Inconsistency of Theory of Nonuniform Torsion:

Overall equilibrium of the bar is satisfied (energy principle). However, only the longitudinal equilibrium equation (along x1) is satisfied locally(St.V. Identical satisfaction of all diff. equil. eqns)

( ) ( ) ( )M M1 1 3 1 1 2 1 1 M

2 2 3 3 1G x x G x x x 0

x x x x xϕ ϕθ θ Ε θ ϕ

↓

⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞∂ ∂∂ ∂ ∂′ ′ ′′⎡ ⎤⋅ ⋅ − + ⋅ + + ⋅ ⋅ =⎢ ⎥⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎣ ⎦∂ ∂ ∂ ∂ ∂⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

( )( )

2 21 1M M

M2 21 12 3

xG xx xΕ θϕ ϕ ϕ

θ′′′⋅∂ ∂

+ = − ⋅′⋅∂ ∂

( ) ( )11 1 1 M 2 3x x , xε θ ϕ′′= ⋅

( )1 2 3M M: x ,x ,xϕ ϕInconsistency Inconsistency

DECOMPOSITION OF DECOMPOSITION OF SHEAR STRESSESSHEAR STRESSES


Physical Meaning of Decomposing Shear StressesPhysical Meaning of Decomposing Shear Stresses

PtM

Angle of Twist Per Unit Length (Torsional

Curvature): ConstantSt.V.

Phase ΙPtM : Primary Twisting

Moment

Pτ

Phase ΙΙWarping Moment

wM :

(A)

σx+dσx

σx

τS1

τS2

τS1 = τS2 = τSCauchy Cauchy

PrinciplePrinciple

StM :

Phase ΙΙΙ

Secondary Twisting Moment


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONPrimary (St. Venant) Shear StressesPrimary (St. Venant) Shear Stresses

Secondary (Warping) Shear StressesSecondary (Warping) Shear Stresses

Normal Normal StressesStresses

( )

( )

PP M12 1 1 3

2

PP M13 1 1 2

3

G x xx

G x xx

ϕτ θ

ϕτ θ

⎛ ⎞∂′= ⋅ ⋅ −⎜ ⎟⎜ ⎟∂⎝ ⎠⎛ ⎞∂′= ⋅ ⋅ +⎜ ⎟⎜ ⎟∂⎝ ⎠

SS M12

2S

S M13

3

Gx

Gx

ϕτ

ϕτ

∂= ⋅

∂

∂= ⋅

∂

( ) ( )w P11 1 1 M 2 3E x x ,xτ θ ϕ′′= ⋅ ⋅

wSS131

PP13 112

2 3 1

11

2 3x x0

xx xττ τττ ∂∂

+∂

∂∂+

∂ ∂∂∂

⎛ ⎞ ⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∂

PP1312

2

2 PM

3x00

xττ ϕ∂∂

+∂ ∂

∇= → =

( )1 12 S P

SS1312

M M

2

w11

13 x

x

0

E

x x

Gθ

ττ

ϕ

τ

ϕ′′′⋅

∇ =

∂∂+

∂ ∂=

⋅

→

−

+∂∂

Reliability of the shear stress decompositionReliability of the shear stress decomposition

( ) ( )S P P1 1 1 1 M 2 3u u x x ,xθ ϕ′= ⋅



α

n

t

12τ

1tτ

13τ

1nτ

Γ Ω

ds

2dx

3dx

Shear stresses along the normal direction nn on

the boundary VANISH:VANISH:

Boundary ConditionsBoundary Conditions

P P P1n 12 2 13 3S S S1n 12 2 13 3

n n 0

n n 0

τ τ τ

τ τ τ

= ⋅ + ⋅ =

= ⋅ + ⋅ =

( )( )

2 2

3 3

n cos x ,n cos

n sin x ,n sin

α

α

= =

= =

( )PM

3 2P

P M2 31n 1 1 3 2 2 3

SS M1

Mn

S

G x x n

0

x n 0 x n x n

0

nn

Gnn

ϕτ θ ϕ

ϕϕτ

∂⎛ ⎞∂′= ⋅ − ⋅ + ⋅ = →⎜ ⎟⎜ ⎟∂⎝ ⎠

∂= ⋅ = →

∂∂

=∂

= ⋅ − ⋅∂

( )P S

P SM M1t 1 1 2 2 3 3 1tG x x n x n G

t tϕ ϕτ θ τ

⎛ ⎞∂ ∂′= ⋅ + ⋅ + ⋅ = ⋅⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONBoundary Value ProblemsBoundary Value Problems

2 P 2 P2 P M M

M 2 22 3

PM

3 2 2 3

0 ,x x

x n x n ,n

ϕ ϕϕ Ω

ϕ Γ

∂ ∂∇ = + =

∂ ∂

∂= ⋅ − ⋅

∂

Primary Warping FunctionLaplace differential eqn with Neumann type boundary conditions

( )2 S 2 S1 12 S PM M

M M2 22 3

SM

x,

Gx x

0 ,n

Ε θϕ ϕϕ ϕ Ω

ϕ Γ

′′′⋅∂ ∂∇ = + = − ⋅

∂ ∂

∂=

∂

Secondary Warping FunctionPoisson differential eqn with Neumann type boundary conditions

(PDEs)


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONStress ResultantsStress Resultants

•• Twisting MomentTwisting Moment: :

•• Primary Twisting MomentPrimary Twisting Moment: :

•• Secondary Twisting MomentSecondary Twisting Moment: :

P S1 1 1M M M= +

( )P P

P P P PM M1 12 3 13 2 1 t 1 1

2 3M x x d M G I x

x xΩ

ϕ ϕτ τ Ω θ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ′⎢ ⎥= − + ⋅ + → = ⋅ ⋅⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∫

P P2 2 M M

t 2 3 2 33 2

I x x x x dx xΩ

ϕ ϕ Ω⎛ ⎞∂ ∂

= + + ⋅ − ⋅⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠∫ Torsion constant Torsion constant

((SaintSaint--VenantVenant))

( )P P

S S S SM M1 12 13 1 M 1 1

2 3M d M E C x

x xΩ

ϕ ϕτ τ Ω θ⎛ ⎞∂ ∂ ′′′= − − → = − ⋅ ⋅⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

∫

( )2PM MC d

Ωϕ Ω= ∫ Warping ConstantWarping Constant

((WagnerWagner))


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONStress ResultantsStress Resultants

•• Warping Warping MomentMoment: :

P wW M 11 W M 1M d M EC

Ωϕ τ Ω θ′′= − → = −∫

Warping Moment as External Warping Moment as External LoadingLoading

• Normal Stresses with Nonuniform Distribution• Bending Moments applied in planes parallel to the longitudinal bar axis located at distance from the center of twist• Concentrated Axial Forces:

( ) ( )1

KP

w Mj jjM P ϕ

=

= −∑e.g. Z-shaped cross section with equal length flanges

at centroid0PMϕ ≠

σxx

Mf

Mf

Mw= Mf h h

Schardt, 1966: “Higher order Stress Resultant”Difficult visualization/depictionSelf-equilibrated stress distribution


Warping Moment due to Axial loading

P

y

z

(Roik, 1978)

P/4

y

z

P/4 P/4

P/4

Axial loading (Ν)

P/4

y

z

P/4 P/4

P/4

Bending loading (Μy)

P/4

y

z

P/4 P/4

P/4

Bending Loading (Μz)

P/4

y

z

P/4 P/4

P/4

Warping Moment Loading (Μw)


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONCenter of TwistCenter of Twist ((ΜΜ))

M M2 3x ,x : Point with respect to which the cross sections rotate (no transverse

displacements) (or point where rotation causes no axial and bending stress resultants

P P12 13 t, ,I :τ τ Independent of the center of twist (St. Venant could not calculate the

position of the center of twist!)P S S w1 12 13 11 Mu , , , ,C :τ τ τ Dependent of the center of twist

( )P P M MM 2 3 O 2 3 2 3 3 2

P2 P O

O 3 2 2 3

x ,x ( x ,x ) x x x x c

0 , x n x n ,n

ϕ φ

ϕϕ Ω Γ

= − + +

∂∇ = = ⋅ − ⋅

∂• Method of equilibrium:

2 3 0N M M= = =Under any coordinate system due to warping normal stresses

• Energy Method:Minimization of Strain Energy due to warping normal stresses

2 30M M MC C C

x x c∂ ∂ ∂

= = =∂ ∂ ∂


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONCenter of TwistCenter of Twist ((ΜΜ))

P2 2 3 3 S

P22 2 23 3 2 2

P23 2 33 3 3 3

S x S x A c R

I x I x S c R

I x I x S c R

Μ Μ

Μ Μ

Μ Μ

− + = −

+ + = −

+ − =

2 3 3 2

2 222 3 33 2 23 2 3

P P P P P PS O 2 3 O 3 2 O

A d S x d S x d

I x d I x d I x x d

R d R x d R x d

Ω Ω Ω

Ω Ω Ω

Ω Ω Ω

Ω Ω Ω

Ω Ω Ω

ϕ Ω ϕ Ω ϕ Ω

= = =

= = = −

= = =

∫ ∫ ∫

∫ ∫ ∫

∫ ∫ ∫

where:


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONGlobal Equilibrium EquationGlobal Equilibrium Equation & Boundary conditions& Boundary conditions

Method of Equilibrium or Energy Method

t 1m dx

M1

11 1

1

MM dx

x∂

+∂

1dx

3x

1x

2x TOTAL POTENTIAL ENERGYL 2

o t 1 1 t 1 10

F

1 1G I E C m dx2 2λ ΜΠ θ θ θ⎛ ⎞′ ′′= ⋅ ⋅ + ⋅ ⋅ − ⋅⎜ ⎟

⎝ ⎠∫

2

21 1 1 11

F d F d F 0dx dx

∂ ∂ ∂∂θ ∂θ ∂θ

− + =′ ′′

(Euler–Lagrange eqns)

t t 1 M 1

1 1 2 1 3 1 1 2 W 3

m G I E Ca a M a M

θ θθ β θ β β

′′ ′′′′= − ⋅ ⋅ + ⋅ ⋅′+ = + =

Inside the bar interval

At the bar ends

Torsional Torsional Damping Damping

CoefficientCoefficient

1515

t

M

GIL :EC

ε≥ →⎧

= ⎨< →⎩

Uniform Torsion

Nonuniform Torsion

(Ramm &Hofmann

1995)


ELASTIC THEORY OF TORSIONELASTIC THEORY OF TORSIONAlternative Solution of the Uniform Torsion ProblemAlternative Solution of the Uniform Torsion Problem

• Conjugate function of function PMϕψ

( )

2 22

2 22 3

2 22 3

0

12

x x

x x Cψ

∂ ψ ∂ ψψ∂ ∂

ψ

⎛ ⎞∇ = + =⎜ ⎟

⎝ ⎠

= ⋅ + +

• Prandtl Stress function F(x,y)

12 3 13 23 2

2 22 3 2 3

2 3

P P

t

G x G xx x

I x x x x dx xΩ

∂ψ ∂ψτ θ τ θ∂ ∂

∂ψ ∂ψ Ω∂ ∂

⎛ ⎞ ⎛ ⎞′ ′= ⋅ − = ⋅ − +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎛ ⎞

= + − ⋅ − ⋅⎜ ⎟⎝ ⎠

∫

2 22

2 22 3

1

F

F FFx x

F C

∂ ∂∂ ∂

⎛ ⎞∇ = + =⎜ ⎟

⎝ ⎠=

12 133 2

2 32 3

2 2P P

t

F FG Gx x

F FI x x dx xΩ

∂ ∂τ θ τ θ∂ ∂

∂ ∂ Ω∂ ∂

′ ′= − ⋅ ⋅ = ⋅ ⋅

⎛ ⎞= ⋅ + ⋅⎜ ⎟

⎝ ⎠∫

Constants Cψ, CF are unknown and must be determined at each boundary of a multiply connected region (occupied by the cross section) Complex Problem.


Example

tw=0.45cm

tf=0.75cm

b=5.5cm

h=10cm

r=1cm

• C M •

eMC

Steel Profile UPE–100 • Bar ends simply supported• Loading: • Length: l=1.0m

1tm kNm m=

Ε=2.1Ε8, ν=0.3

FΕΜ (Kraus, 2005) ΒΕΜ

Pmφ P

mφ

Warping along the thickness direction IS NOT IS NOT

CONSTANTCONSTANT Thin walled beam theory not valid

( )0 0 1 0x . , . m=( ) 0 024P

xmax u cm .=( )0 0 1 0x . , . m=

( ) 44 052 10Sxmax u cm . −= ×

ΒΕΜ

0 016S Px xmaxu . maxu=

( )wσ→

3 66 15.ε = <


Example

Bar of Box Shaped Cross Section Clamped at Both

Ends

P=100kN

3.65m

l/2=20m

0.3

0

0.3

0

0.4

5

0.4

5

0.7

5

0.3

0

3.4

5

0.6

0

7.60m

4.50

3.65

3.30

2.50

2.00

3.50 3.75

z

y Α

Β

(E=3.0Ε7, G=1.5E7)

Primary Warping

PMφ

0 10 20 30 405 15 25 35

600

400

200

0

-200

-400

-600

500

300

100

-100

-300

-500

My (kNm): Καμπτική Ροπή (min/max: 500)Mw (kNm2): Διρροπή (min:-343.00, max:+333.98)

Bar Length (m)

23.32>15ε =

0 10 20 30 405 15 25 35

80

40

0

-40

-80

60

20

-20

-60

σb: Κάμψη (θλ:-49.4765, εφ:+49.4765)σw: Διρροπή (θλ:-21.6758, εφ:+22.2612)σtot: Κάμψη & Διρροπή (θλ:-71.1523, εφ:+71.7377)

0 10 20 30 405 15 25 35

80

40

0

-40

-80

60

20

-20

-60

σb: Κάμψη (θλ:-40.9965, εφ:+40.9965)σw: Διρροπή (θλ:-26.7831, εφ:+26.0788)σtot: Κάμψη & Διρροπή (θλ:-67.7796, εφ:+67.0753)

Bar Length (m) Bar Length (m)

Point Α

Point Β

NORMAL STRESSES

Discrepancy30%≅


Example

Bar of Box Shaped Cross Section Clamped at Both

Ends

P=100kN

3.65m

l/2=20m

0 10 20 30 405 15 25 35

-200

-100

0

100

200

-150

-50

50

150

Mtp: Πρωτογενής Στρεπτική Ροπή (min/max: 180.72)Mts: Δευτερογενής Στρεπτική Ροπή (min/max: 182.50)Mt: Συνολική Στρεπτική Ροπή (min/max: 182.50)

Bar length (m)

23.32>15ε =

SHEAR STRESSES

Discrepancy30%≅


α α

Β2

Δ2

Γ

b(y)

Qz

z

y C

(C: Centroid)

Β1 Β3

Δ1 Δ3

0

cutz y

xzyyz

xy

Q SI bQ :

τ

τ

⎧=⎪

⎨⎪ =⎩

UNIFORM SHEAR BEAM THEORYUNIFORM SHEAR BEAM THEORY• Computation of Shear Stresses

• Computation of Shear Center Position

• Computation of Shear Deformation Coefficients (required for Timoshenko beam theory)

Cross sections possessing at least one axis of symmetrySimply connected cross section

Poisson ratio ν neglectedτxz: constant along the width b

Β1Β3 & Δ1Δ3 τxz: vanishing

Β2Δ2 point Γ τxz: Discontinuity

BVPBVPDisplacement Field

Stress Field Poisson ratio ν taken into accountTWBTTWBTShear StressesShear Stresses

GBTGBT

ΤΤBTBT


ASSUMPTIONS OF ELASTIC THEORY OF UNIFORM SHEAR

•• The bar is straightThe bar is straight..•• The bar is prismaticThe bar is prismatic..•• Distortional deformations of the cross section are not allowed (Distortional deformations of the cross section are not allowed (γγ23=0, distortion neglected). 23=0, distortion neglected).

•• The material of the bar is homogeneous, isotropic, continuous (nThe material of the bar is homogeneous, isotropic, continuous (no cracking) and o cracking) and linearly elastic: Constitutive relations of linear elasticity arlinearly elastic: Constitutive relations of linear elasticity are valide valid..•• The distribution of stresses at the bar ends is such so that allThe distribution of stresses at the bar ends is such so that all the aforementioned the aforementioned assumptions are validassumptions are valid..

•• Longitudinal displacements Longitudinal displacements (warping) are not restrained and do (warping) are not restrained and do not depend on the longitudinal not depend on the longitudinal coordinate (every cross section coordinate (every cross section exhibits the same warping exhibits the same warping deformations)deformations). .

•• Deflections and bending rotations Deflections and bending rotations are considered to be small are considered to be small (geometrically linear theory)(geometrically linear theory). .


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEAR –– DISPLACEMENT FIELDDISPLACEMENT FIELD

Displacement fieldDisplacement field

( ) ( ) ( ) ( )( ) ( )( ) ( )

1 1 2 3 2 1 3 3 1 2

2 1 2 3 2

c 2 3

1

3 1 2 3 3 1

x ,xu x ,x ,x x x x x

u x ,x ,x u x

u x ,x ,x u x

θ θ ϕ= ⋅ − +

=

=

2x3x

1x

• • S

• S •

2Q

3Q

3Q

2M3M

• S

C

C

2Q

l

x1

l-x1

By ignoring Shear Strains:

( ) ( )3 22 1 3 1

1 1

u ux xx x

θ θ∂ ∂= − =

∂ ∂



Components of the Infinitesimal Strain TensorComponents of the Infinitesimal Strain Tensor

3 31 2 211 3 2 22 33

1 1 1 2 3

uu ux x 0 0x x x x x

θθε ε ε∂ ∂∂ ∂ ∂= = − = = = =∂ ∂ ∂ ∂ ∂

c c1 2 212 3

2 1 1 2 2

u u u1 1 12 x x 2 x x 2 x

ϕ ϕε θ⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂

= + = − + + =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

3223

3 2

uu1 02 x x

ε⎛ ⎞∂∂

= + =⎜ ⎟∂ ∂⎝ ⎠

3 c 3 c113 2

3 1 3 1 3

u uu1 1 12 x x 2 x x 2 x

ϕ ϕε θ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂∂

= + = + + =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠



Components of the Cauchy Stress TensorComponents of the Cauchy Stress Tensor ((νν=0=0))

3211 11 3 2

1 1E E x x

x xθθτ ε

⎛ ⎞∂∂= ⋅ = −⎜ ⎟∂ ∂⎝ ⎠

22 33 230 0 0τ τ τ= = =

c12 12

22G G

xϕτ ε ∂

= ⋅ =∂

c13 13

32G G

xϕτ ε ∂

= ⋅ =∂


( )2 2

2 3 3 2 2 22

3111 21

1 2 3

3212 22 12

1 2 3 1

13 23 33 13

1 2 3 1

30

0:

0

0

0

0

c cE xx x x

x x x x

x x

G

x

xx x

x

Gϕ ϕθττ τ

ττ τ τ

τ τ τ τ

θ∂∂ ∂+ + = →

∂ ∂ ∂

∂∂ ∂ ∂ ⎫+

∂ ∂′′ ′′⋅ − ⋅ + + =∂

+ = =→ ⎪∂ ∂ ∂ ∂ ⎪⎬∂ ∂ ∂ ∂ ⎪+ + = → =⎪∂

∂

∂ ∂ ∂ ⎭

Identical Satisfaction



Determination ofDetermination of:: ( )2 3 3 2E x xθ θ′′ ′′⋅ − ⋅

( )

( )

22 11 3 2 3 3 2 3 2 22 3 23

23 11 2 2 2 3 3 2 3 33 2 23

M x d E x d x x d E I I

M x d E x x d x d E I I

Ω Ω Ω

Ω Ω Ω

τ Ω θ Ω θ Ω θ θ

τ Ω θ Ω θ Ω θ θ

⎫⎛ ⎞′ ′ ′ ′= ⋅ = − = ⋅ − ⋅ ⎪⎜ ⎟⎜ ⎟ ⎪⎝ ⎠

⎬⎛ ⎞ ⎪′ ′ ′ ′= − ⋅ = − − = ⋅ − ⋅⎜ ⎟ ⎪⎜ ⎟⎝ ⎠ ⎭

∫ ∫ ∫

∫ ∫ ∫

Ι22, Ι33, Ι23: Moments of inertia

( )

( )

23 2 22 3 23

1

32 2 23 3 33

1

MQ E I IxMQ E I Ix

θ θ

θ θ

∂ ⎫′′ ′′= = ⋅ − ⋅ ⎪∂ ⎪→⎬∂ ⎪′′ ′′= − = ⋅ − ⋅⎪∂ ⎭

( ) ( ) ( )3 33 2 23 3 2 22 3 23 22 3 3 2 2

22 33 23

Q I Q I x Q I Q I xE x x

I I Iθ θ

− + −′′ ′′− =

−

Equilibrium ofbending moments



Poisson Partial Differential EquationPoisson Partial Differential Equation

( ) ( )2 2

22 3 2 32 2

2 3, , ,c c

c x x f x xx xϕ ϕϕ ∂ ∂

∇ = + =∂ ∂

Ω

( ) ( )3 32 21, ,gfG

x x xx −=

( ) ( ) ( )2 3 3 33 2 23 3 2 22 3 23 21,g x x Q I Q I x Q I Q I xD⎡ ⎤= − + −⎣ ⎦

222 33 23D I I I= −

Boundary ConditionBoundary Condition

α

n

t

12τ

1tτ

13τ

1nτ

Γ Ω

ds

2dx

3dx

1n 12 2 13 3

c c c2 3

2 3

c

n n 0

G G n G n 0n x x

,0n

τ τ τϕ ϕ ϕ

Γϕ

= + = →

∂ ∂ ∂= +

∂

→

∂

=

=

∂ ∂ ∂

(Neumann)


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEAR –– STRESS FIELDSTRESS FIELD

11 2 23 3 22 2 33 3 232 32 2

22 33 23 22 33 23

Μ Ι +Μ Ι Μ Ι +Μ Ι& x + xΙ Ι - I Ι Ι - I

τ⎛ ⎞ ⎛ ⎞

=−⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

22 33 23

323 2

1

2 3

1

0

(statically determinate beam

are computed through the global equilibrium equati n )o s

MMQ ,Qx

Mx

,M

τ τ τ= = =∂∂

= = −∂ ∂

⇒

Beam theory:


( )

:

3111 21

1 2 3

3212 22 12

1

31 3212 23 3 3

2 3 1

13 23 3

322 3 22 33

3 13

1 2 3 1

230

x x x

0 0x x x x

0

Q x I x Ix x I

0x x x x

I Iττ τ

ττ τ τ

τ τ τ

ττ

τ

∂∂ ∂+ + = →

∂ ∂ ∂

∂∂ ∂ ∂ ⎫+ + = → = ⎪∂ ∂ ∂ ∂ ⎪⎬∂ ∂ ∂ ∂ ⎪+ + = → =⎪∂ ∂

∂∂+ = −

∂ ∂

∂ ∂ ⎭

−

•• AnalysisAnalysis:: QQ33

Shear stresses depend on x2 & x3, exclusively, that is they are the sameat each cross section of the bar

(Q2 correspondingly and subsequent superposition of results)


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEAR –– STRESS FIELDSTRESS FIELD

1111 E

τε =

( )v 0≠Components of the infinitesimal strain tensorComponents of the infinitesimal strain tensor

22 33 11 11Eνε ε τ νε= = − = −

( )1212 12 2 3x ,x

2Gτε ε= =

23 0ε =

( )1313 13 2 3x ,x

2Gτε ε= =

((ANALYSIS:ANALYSIS: QQ33))

Compatibility EquationsCompatibility EquationsStrain field Satisfies 4 compatibility equations identically

( )( )

( )( )

13 3 33122

2 2 3 22 33 23

13 3 23122

3 2 3 22 33 23

Q Ix x x 1 I I I

Q Ix x x 1 I I I

τ ντ

ν

τ ντ

ν

⎛ ⎞∂ ∂∂− =⎜ ⎟∂ ∂ ∂ + −⎝ ⎠

⎛ ⎞∂ ∂∂− =⎜ ⎟∂ ∂ ∂ + −⎝ ⎠


Stress FunctionStress Function

Shear stressesShear stresses:: They satisfy Compatibility Equations Identically

3 312 2 13 3

2 3

Q Qd dB x B x

Φ Φτ τ⎛ ⎞⎛ ⎞∂ ∂

= − = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

( )2 3x ,x :Φ Stress function with continuous partial derivatives up to 2nd order

2 3

2 3

d i i

i i

2 3

2 2 2 22 3 2 3

33 2 3 23 33 23 2 3

d d

x x x xI x x I I I x x2 2

ν ν

= + =

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− −− + − +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

( )( )222 33 23B 2 1 I I Iν= + −

THEORY OF UNIFORM SHEAR THEORY OF UNIFORM SHEAR –– STRESS FIELDSTRESS FIELD ((AnalysisAnalysis:: QQ33))


Poisson Partial Differential EquationPoisson Partial Differential Equation

( ) ( )2 2

22 3 2 23 3 332 2

2 3, 2 ,x x x I x I

x x∂ Φ ∂ Φ

∇ Φ = + = −∂ ∂

Ω


α

n

t

12τ

1tτ

13τ

1nτ

Γ Ω

ds

2dx

3dx

n d

1n 12 2 13

2 3

3

2 3d n d n

n n 0

n,Φ

τ τ τ

Γ∂= + =

+ =

⋅∂

= →


(Neumann)



Poisson Partial Differential Poisson Partial Differential EquationEquation

( )223 3 22 22 ,I x I x∇ Θ = − Ω


n e2 2 3 3e n e n ,nΘ Γ∂

= + = ⋅∂

2 3

2 3

e i i

i i

2 3

2 2 2 22 3 2 3

22 23 2 3 23 22 2 3

e e

x x x xI I x x I I x x2 2

ν ν

= + =

⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞− −− + +⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

(Neumann)


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEAR

Shear CenterShear Center ((SS))• Point where Internal Shear Stress Resultant is subjected

• Poisson ratio ν = 0 S.C. (S) coincides with C. of T. (Μ) (Weber, 1924), (Trefftz, 1935)

• Determination: With respect to an arbitrary point

Μtext : Twisting Moment at S.C. arising from externally applied forces

Μtint : Twisting Moment arising from shear stresses due to transverse shear

Μtext = Μt

int

( )

ExtS S

3 2 2 3Int

13 2 12 3

Q x Q x

x x dΩτ τ Ω

⋅ − ⋅ =

− →∫

S

(Ω)

n

s

K 1j 1 jΓ Γ+==∪ t

C (G,v) x2

x3

s Γ1

Γ2

ΓK

ΓK+1



SHEAR CENTERSHEAR CENTER –– DISPLACEMENT FIELDDISPLACEMENT FIELD

2 2

3 3

cx cxS2 2 3

3 2

cx c

2 3

xS3 22

3 23 3

x G x x dx x

x G x x dx x

Q

Q 0 &

1&Q 0 :

Q 1 : Ω

Ω

ϕ ϕΩ

ϕ ϕΩ•

∂ ∂⎛ ⎞= −⎜ ⎟

∂ ∂

= =

⎝ ⎠∂ ∂⎛ ⎞

= − −⎜ ⎟∂

• =

∂⎝ ⎠

= ∫

∫

SHEAR CENTERSHEAR CENTER –– STRESS FIELDSTRESS FIELD

S2 2 3 2 3 3 2

3 2

S3 3 2 32 23

2

3

3

22 3

1x x x x d x d dB x x

1x x x x e x e dB x

Q 0 &Q 1:

Qx

1&Q 0 :

Ω

Ω

Φ Φ Ω

Θ Θ Ω•

⎡ ⎤∂ ∂= − − +⎢ ⎥∂ ∂⎣ ⎦

⎡ ⎤∂ ∂= − − +⎢ ⎥∂ ∂

=

• =

⎣ ⎦=

= ∫

∫



Q3

( ) ( )13 1 31 313 : Actual shear strains

x ,xx ,x

Gτ

γ =

Shear Deformation Shear Deformation CoefficientCoefficient αα33 (>1)

13 3 13aγ γ=Shear Correction FactorShear Correction Factor

κκ33 (<1)

13 13 3 133

1a

γ γ κ γ= =3 33

1 :sA A Aa

κ= = Effective Shear Area

( )( )1 313

13 1 3 : Average shear strainsx ,x

x ,xA

Ω

γγ =

∫

13: Shear strains of Timoshenko beam theory They need correction since theyhave unsatisfactory (constant) distribution

γ⇒

133 3 13 3sQ G A GAγκ γ==


THEORIES OF SHEAR DEFORMATION COEFFICIENTSTHEORIES OF SHEAR DEFORMATION COEFFICIENTS

1) Timoshenko Theory (1921, 1922):

2) Cowper Theory (1966):

3) Energy Approach (Bach & Baumann, 1924):

3κ =Average value of shear stresses

Actual shear stress at centroid

(if centroid does not lie in the cross section ?)

Global equilibrium equations formulated by integrating the 3d elasticity differential equilibrium equations

The formulas of the approximate shear strain energy per unit length and the exact one are equated

αα33 must depend on the ratio of the sides (must depend on the ratio of the sides (b/hb/h))

h ↓ b h ↑ 3 31 0aκ = →that is then so that

Unacceptably large values

!!!

3

22

sGAEI

→

13 3 13133

1 0a

γ γ κ γ= = →

IfIf αα33 is independent ofis independent of b/hb/h thenthen⎧⎨⎩

Unrealistic results

FEM: “Shear–Locking”


≠Principal Shear Principal Shear

SystemSystemPrincipal Bending Principal Bending

SystemSystem

23

22 33

22 B ItanI I

ϕ =−

23

2 3

22 S atana a

ϕ =−

Bending Deflections: COUPLEDCOUPLED

Axis of symmetry S Bϕ ϕ=

Shear Deformation CoefficientsShear Deformation Coefficients

Exact formula of shear strain energy per unit length =Approximate formula of shear strain energy per unit length

apprexact UU

2 2 2212 13 3 3 23 2 32 2 Q Q QQd

2G 2 G 2 G GΩ

τ τ α ααΩΑ Α Α

+= + +∫


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEARSHEAR DEFORMATION COEFFICIENTSSHEAR DEFORMATION COEFFICIENTS

DISPLACEMENT FIELDDISPLACEMENT FIELD

2 3

222c2 c2

2 22 32

2 3

222c3 c3

3 22 33

2 3

222 2c23 c23 c3

23 2 22 3

Q 0, Q 0 :

AGa dx xQ

Q 0, Q 0 :

AGa dx xQ

Q 0, Q 0 :

AG AGa dx x xQ Q

Ω

Ω

Ω

ϕ ϕ Ω

ϕ ϕ Ω

ϕ ϕ ϕΩ

≠ =

⎡ ⎤⎛ ⎞⎛ ⎞∂ ∂⎢ ⎥= + ⎜ ⎟⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦= ≠

⎡ ⎤⎛ ⎞⎛ ⎞∂ ∂⎢ ⎥= + ⎜ ⎟⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦≠ ≠

⎡ ⎤⎛ ⎞⎛ ⎞∂ ∂ ∂⎢ ⎥= + −⎜ ⎟⎜ ⎟∂ ∂ ∂⎢ ⎥⎝ ⎠

•

•

•

⎝ ⎠⎣ ⎦

∫∫

∫∫

∫∫22

c3

2 3

222c2 c2

22 3

dx

AG dx xQ

Ω

Ω

ϕ Ω

ϕ ϕ Ω

⎡ ⎤⎛ ⎞⎛ ⎞ ∂⎢ ⎥+ ⎜ ⎟⎜ ⎟ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞⎛ ⎞∂ ∂⎢ ⎥− + ⎜ ⎟⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∫∫

∫∫


THEORY OF UNIFORM SHEARTHEORY OF UNIFORM SHEARSHEAR DEFORMATION COEFFICIENTSSHEAR DEFORMATION COEFFICIENTS

STRESS FIELDSTRESS FIELD

( ) ( )

( ) ( )

( ) ( )

Θ e Θ e

Φ d Φ d

Φ d Θ e

2 3

2 2

2 3

3 2

2 3

23 2

Q 0, Q 0 :

dB

Q 0, Q 0 :

dB

Q 0, Q 0 :

2 dB

Ω

Ω

Ω

Αα Ω

Αα Ω

Αα Ω

≠ =

= ∇ − ⋅ ∇ −

= ≠

= ∇ − ⋅ ∇ −

≠ ≠

⎡ ⎤= ∇ − ⋅ ∇ −⎣

•

⎦

•

•

∫

∫

∫


Example Rectangular section:

b

h

Qz=1kNbxh=60x30 (cm)

8.33kPa

-0.2 0.0 0.2-0.3 -0.1 0.1 0.3

6

8

10

12

14

7

9

11

13

Λόγος Poisson v=0.0Λόγος Poisson v=0.3

y y

Centroidal Axis yy

Shear stresses τΩ (kPa)

Discrepancy31%≈

ΤBT:

-0.30 -0.20 -0.10 -0.00 0.10 0.20 0.30-0.15

-0.05

0.05

0.15

-0.30 -0.20 -0.10 -0.00 0.10 0.20 0.30-0.15

-0.05

0.05

0.15

12.14kPa

2.94kPa

Poisson ratio:ν=0.0

1 5 8 33max zxz

Q. . kPaA

τ = =

Poisson ratio:ν=0.3


Example

b=30cmh

“Shear Locking”(Artificial, Spurious

(Shear) Rigidity induced)

sz yyGA EI

2 4 6 8 101 3 5 7 9

0

1000

2000

3000

4000

5000

500

1500

2500

3500

4500

Ενεργειακή Μέθοδος (Προτεινόμενη)

Timoshenko

Cowper

b=30cm

h

Τεχνική θεωρία

b h


Example

C

b

b=0.10m

A B

bbb

b

b

b

z

y

0.16

67m

Γ

Δ

T-shaped cross section

0.002.004.006.008.0010.0012.0014.0016.0018.0020.0022.0024.0026.0028.00

10 3

zQ kNv .= −=

0.002.004.006.008.0010.0012.00

14.0016.0018.0020.00

1

0 3yQ kN

v .

=

=

Poisson ratio νDoes not affect shear

center-0.20 -0.10 0.00 0.10 0.20-0.15 -0.05 0.05 0.15

5

6

7

8

9

10

5.5

6.5

7.5

8.5

9.5

Λόγος Poisson v=0.0Λόγος Poisson v=0.3Τεχνική θεωρία δοκού

1zQ kN= −

Segment ΑΒ

τΩ (kPa)Segment ΓΔ

0.0 4.0 8.0 12.0 16.0 20.02.0 6.0 10.0 14.0 18.0

0

0.1

0.2

0.3

0.4

0.05

0.15

0.25

0.35

Λόγος Poisson v=0.0Λόγος Poisson v=0.3Τεχνική θεωρία δοκού

τΩ (kPa)

1zQ kN= −


Example Thin walled L-shaped cross section

C

z

y

b=10.5cm

h=15

.5cm

t=1cm

t=1cm

3.995cm

1.49

5cm

φB=0.430249(rad)

S φS

Principal Bending

Principal Shear

8.6c

m

2kNv=0.3:max 0.09207cmΓτ =

2kNv=0.0:max 0.09093cmΓτ =

2kNv=0.3:max 0.13605cmΓτ =

2kNv=0.0:max 0.13212cmΓτ =

4.1cm

1zQ kN= −

1yQ kN=

Small influence of Poisson ratio νin stresses and Shear Center at thin

walled cross section bars


ASSUMPTIONS OF TIMOSHENKO BEAM THEORY

•• The bar is straightThe bar is straight..•• The bar is prismaticThe bar is prismatic..•• Distortional deformations of the cross section are not allowed (Distortional deformations of the cross section are not allowed (γγyzyz=0, distortion neglected). =0, distortion neglected).

•• The material of the bar is homogeneous, isotropic, continuous (nThe material of the bar is homogeneous, isotropic, continuous (no cracking) and o cracking) and linearly elastic: Constitutive relations of linear elasticity arlinearly elastic: Constitutive relations of linear elasticity are valide valid..•• External transverse forces pass through the cross sectionExternal transverse forces pass through the cross section’’s shear center. Torsional and s shear center. Torsional and axial forces are not considered (axial forces are not considered (torsionlesstorsionless bending loading conditions)bending loading conditions)..•• Deflections and bending rotations Deflections and bending rotations are considered to be small are considered to be small (geometrically linear theory)(geometrically linear theory). .

•• The distribution of stresses at the The distribution of stresses at the bar ends is such so that all the bar ends is such so that all the aforementioned assumptions are aforementioned assumptions are valid.valid.

•• Cross sections remain plane after Cross sections remain plane after deformationdeformation.. X

y

zZ

Y

L

S

Cx

C: centroid S: shear center

ZpYp

Ym

Zm

Ym

Yp

Zm

Zp

SC

dx


TIMOSHENKO BEAM THEORYTIMOSHENKO BEAM THEORY

Displacement Field:Displacement Field: (Arising from the plane sections hypothesis)

α

ω ΓΚ

r q P= −

q

PS

Kjj 0Γ Γ= ∪ =

,y v

t

n

s

C

,z w,Z W

1Γ,Y V

0Γ

zC

yC

CXYZ CUse of the principal passing through the centroid shear system

X

y

zZ

Y

L

S

Cx

C: centroid S: shear center

ZpYp

Ym

Zm

Ym

Yp

Zm

Zp

SC

dx

( ) ( )( ) ( )( ) ( ) ( )Y C Z Y ZCu x,y,z x z z x xy Yxy Zθ θθ θ= − − − = −

( ) ( )v x,z v x=

( ) ( )w x, y w x=

( ) ( )Y x w xθ ′≠ −

( ) ( )Z x v xθ ′≠



Components of the Infinitesimal Strain Tensor Components of the Infinitesimal Strain Tensor (Geometrically linear theory)(Geometrically linear theory)

Y Zxx

d du Z Yx dx dx

θ θε ∂= = −∂

yyv 0y

ε ∂= =∂ zz

w 0z

ε ∂= =∂

xy Zu v dvy x dx

γ θ∂ ∂= + = −∂ ∂

yzv w 0z y

γ ∂ ∂= + =∂ ∂

xz Yu w dwz x dx

γ θ∂ ∂= + = +∂ ∂



Components of the Cauchy Stress TensorComponents of the Cauchy Stress Tensor ((νν=0)=0)

( )( ) ( ) ( )xx xx yy zzY ZE d dE Z Y

dx d1 1 2 x1σ ν ε ν ε ε

ν νθ θ⎡ ⎤= − + + =⎣ ⎦

⎛ ⎞−⎜ ⎟⎝− ⎠+

( )( ) ( ) ( )yy yy xx zzE 1 0

1 1 2σ ν ε ν ε ε

ν ν⎡ ⎤= − + + =⎣ ⎦+ −

( )( ) ( ) ( )zz zz xx yyE 1 v v 0

1 v 1 2vσ ε ε ε⎡ ⎤= − + + =⎣ ⎦+ −

( )xy xy ZGG vθτ γ ′−⋅ += =

yz yzG 0τ γ= ⋅ =

( )xz xz YGG wθτ γ⋅ = ′+=




Not Satisfied!Inconsistency of Timoshenko Beam Theory:

Overall equilibrium of the bar is satisfied (energy principle). The violation of the longitudinal equilibrium equation (along x) and of the associated boundary condition is due to the unsatisfactory distribution of the shear stresses arising from the plane sections hypothesis. Thus, in order to correct at the global level this unsatisfactory distribution of shear stresses, we introduce shear correction factors in the cross sectional shear rigidities at the global equilibrium equations

( )xy xy ZGG vθτ γ ′−⋅ += = ( )xz xz YGG wθτ γ⋅ = ′+=

constant distributi unsatisfacon: tory


TIMOSHENKO BEAM THEORYTIMOSHENKO BEAM THEORYStress ResultantsStress Resultants

•• Shear stress resultantsShear stress resultants: : yy xy ZdvQ d Gdx

AΩτ Ω θ⎛ ⎞= = −⎜ ⎟

⎝ ⎠∫

zz xz YdwQ d Gdx

AΩτ Ω θ⎛ ⎞= = +⎜ ⎟

⎝ ⎠∫

: Shear areas with respect

to the y,z axesy zA , A

yyy

1A A Aa

κ= = zzz

1A A Aa

κ= =

: shear correction factors <( )1y z,κ κ: shear deformation coefficients( )>1y za ,a

From the assumed displacement field we would have obtained shear rigidities GA which are larger than the actual ones⎛ ⎞⎜ ⎟⎝ ⎠Since we are working with the principal shear system of axes

Thus the relations of shear stress resultants with respect to the kinematicalcomponents are decoupled

yza 0=⇒



yy xy ZdvQ d Gdx

AΩτ Ω θ⎛ ⎞= = −⎜ ⎟

⎝ ⎠∫

zz xz YdwQ d Gdx

AΩτ Ω θ⎛ ⎞= = +⎜ ⎟

⎝ ⎠∫

yyy

1AA Aa

κ= = zzz

1AA Aa

κ= =

In general we would have :yza 0≠

yzyz

yz1A AA

aκ

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= =

exac appt rU

2 2y y yz y z

U

2z z

2xy xz Q Q QQ

2AG 2AG AGd

2GΩ

α αατ τΩ +

+= +∫

From this theoryFrom uniform shearbeam theory



•• Bending momentsBending moments: : Y xxM ZdΩσ Ω= ∫

Bending moments are defined with respect to the principal shear systemof axes passing through the of the cross c sentroid ection

Z xxM YdΩσ Ω= − ∫

Y ZY Y

2xx Z

Y ZY

d dM Zd EZ d EYZ ddx d

d dEI EIdx dx xΩ Ω Ω

θ θσ Ω Ω Ω θ θ= = − = −∫ ∫ ∫

Z YZ

2 Zxx Z

YZ Y

d dM Yd EY d EYZ d dEI Eddx d

Idx dx xΩ Ω Ω

θ θθ θσ Ω Ω Ω −= − = − =∫ ∫ ∫

:

Moments of inertia with respect to the centroid of the cross section

2 2Y Z YZI Z d , I Y d , I YZd

Ω Ω ΩΩ Ω Ω= = =∫ ∫ ∫


TIMOSHENKO BEAM THEORYTIMOSHENKO BEAM THEORYGlobal Equilibrium EquationsGlobal Equilibrium Equations & Boundary conditions& Boundary conditions

Method of Equilibrium or Energy Method

TOTAL POTENTIAL ENERGY2

21 1 1 11

F d F d F 0dx dx

∂ ∂ ∂∂θ ∂θ ∂θ

− + =′ ′′

(Euler–Lagrange eqns)

S

x ,u

z,w Qz

MY Qz+dQz

MY+dMY pz y,v w–dw

w

–dw

dx x

mY C

x ,u

y,v

Qy

MZ Qy+dQy

MZ+dMZ py z,w v–dv

v –dv

dx x

mZ C

S



Method of Equilibrium

S

x ,u

z,w Qz

MY Qz+dQz

MY+dMY pz y,v w–dw

w

–dw

dx x

mY C

x ,u

y,v

Qy

MZ Qy+dQy

MZ+dMZ py z,w v–dv

v –dv

dx x

mZ C

S

Equilibrium of bending moments

Yz Y

dM Q m 0dx

− + =

Zy Z

dM Q m 0dx

+ + =

Equilibrium of transverse shear forces

yY

dQp 0

dx+ =

zZ

dQ p 0dx

+ =



Equilibrium of bending moments

( )2 2

Y ZY YZ Y Y2z 2

Z

YY

dM Q m d0dx

d GA dwEI EI m 0 1a dxdx dx

θ θ θ⎛ ⎞− − + + =⎜ ⎟⎠

⇒⎝

− + =

( )2 2

Z YZ YZ Z Z2y 2

Y

ZZ

dM Q m d0dx

d GA dvEI EI m 0 2a dxdx dx

θ θ θ⎛ ⎞− + − + =⎜ ⎟⎠

⇒⎝

+ + =

Equilibrium of transverse shear forces

( )2

Zy2y

Y

y dGA d v p 0 3dQ

p 0dx a dxdx

θ⎛ ⎞− + =⎜ ⎟⎜ ⎟

⎝ ⎠+ = ⇒

( )2

Yzz2

Zz

dGA ddQ p 0d

w p 0 4a dx dxx

θ⎛ ⎞+ + =⎜ ⎟⎜ ⎟

⎝ ⎠+ = ⇒

Inside the bar interval

1 2 y 3v Qβ β β+ = 1 2 z 3w Qγ γ γ+ =

1 Z 2 Z 3β θ β Μ β+ = 1 Y 2 Y 3γ θ γ Μ γ+ =At the bar ends

Coupled system of equations due to principal shear system of axes and due toshear deformation effects→



Combination of equations may be performed in order to uncouple the problem unknowns - Solution with respect to bending rotations (or deflections)

( ) ( ) ( ), into 3 3

Y Z YY YZ y3 3

d d dmEI EI p 0 1' 1 3dxdx dx

θ θ− + + =

( ) ( ) ( ), into 3 3

Z Y ZZ YZ z2 2

d d dmEI EI p 0 2' 2 4dxdx dx

θ θ− + − =

1 Z 2 Z 3β θ β Μ β+ =

1 Y 2 Y 3γ θ γ Μ γ+ =

( )2 2

Y ZY YZ Y Y2 2

Z

d d GA dwEI EI m 0 1a dxdx dx

θ θ θ⎛ ⎞− − + + =⎜ ⎟⎝ ⎠

( )2 2

Z YZ YZ Z Z2 2

Y

d d GA dvEI EI m 0 2a dxdx dx

θ θ θ⎛ ⎞− + − + =⎜ ⎟⎝ ⎠

1 2 y 3v Qβ β β+ =

1 2 z 3w Qγ γ γ+ =

Resolution of rotations:

Resolution of deflections:


TIMOSHENKO BEAM THEORY TIMOSHENKO BEAM THEORY -- EXAMPLEEXAMPLE

L x3

P3

A L x1

O

a bFind the elastic curve and the reactions of the beam using Timoshenko beam theory

Torsionless bending on the Ox1x3 plane

( )

( )

( )

: Equation of equilibrium (forces):

(a)

(b)

STEP 1 33 3

32 2 22 23 2 23 3 3

122

11 1 1

2 1211

222

2 1211

dd EI =0, 0 x adxdx

1'

dd dm dEI EI p 0 EI 0dx

dd EI =0, a x

dx dx d

dxx

x

Ld

θ

θθ θ

θ

⎛ ⎞⎜ ⎟ < <⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟ < <⎜ ⎟

− + +

⎝ ⎠

= ⇒ = ⇒



L x3

P3

A L x1

O

a bFind the elastic curve and the reactions of the beam using Timoshenko beam theory

Torsionless bending to the Ox1x3 plane

( )( )

( )( )

( )( )

( )( )

( )

Integrate three times eqns (a,b):

(c)

(d)

1 11 2

1 22 22 1 2 23 3

1 1 1 1

1 21 22 2

2 1 1 3 2 2 1 42 21 1

21 1

2 1 3 1 52

0 x a a x L

d dd dQ = EI =C Q = EI =Cdx dx dx dx

d dM = EI =C x +C M = EI =C x +C

dx dx

xEI =C +C x +C2

θ θ

θ θ

θ

< < < <

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

( ) (e)2

2 12 2 4 1 62

xEI =C +C x +C2

θ



( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 6relations (e)

relatio

: Resolve constants C -C by exploiting the boundary conditions (rotations, moments):

at : (f)

at :

STEP 21 1

1 52 22 2

12 2

1. x1 x 0 0 0 C 0

2. M x1 x L M L =0

θ θ= = ⎯⎯⎯⎯⎯→ =

=( ) ( ) ( ) ( )

ns (d)

relations (e)

(g)

Rotational continuity condition at : (h)

(i)

Equilibrium con

2 4 51 2

1 2 222

21 3 4 6

C L+C C 0

3. x a a = a

C aaC +C a= +C a+C2 2

4.

θ θ

⎯⎯⎯⎯⎯→ =

=

⎯⎯⎯⎯⎯→

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

ditions (forces, moments) at :

(j)

11 2- +

33 3

1 22 2

x a

Q a =Q a +P

M a = M a

=

relations (c,d)

(k)

(l)

1 2 3

1 3 2 4

C C PC a C C a C

⎯⎯⎯⎯⎯→ = +

+ = + (m)From relations (g), (i), (l), (m)

(n)21 2 3 3 2 3 4 2 6 3C C P C C L aP C C L C a P / 2

⇒

= + = − − = = −



( )( )

( ) ( )( )

: Resolve deflections from equation of equilibrium (moments):(1P )STE 322 2

3 3 32 22 23 2 2 2 3 22

1 11 13 2

3 1 2

2 23 1 11 1

2 31 1

1

1

d du dud dGAEI EI m 0 EI k GA 0a dx dxd

du ddk GA Q x EId

x dx

x x dx

x

d

dθθ θθ θ

θθ

⎛ ⎞ ⎛

⎛ ⎞ ⎛⎜ ⎟+ = =

⎞− − + + = ⇒ − + = ⇒⎜ ⎟ ⎜ ⎟

⎠

⎟

⎝

⎜

⎝ ⎠

⎝ ⎠

( )( )

( ) ( )( )

(o)

(o)

1

2 22 23 2

3 1 2 12 31 1 1

, 0 x a

du ddk GA Q x EI , a x Ldx dx dx

θθ

⎞⎜ ⎟ < <⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+ = = < <

⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( ) ( ) ( )

( )

Substitute the values of constants (f), (n) into relations (e) and (c) and theresulting expressions in (o), we get:

(p)21

2 3 1 2 3 13 2 3

1 3 222 2

3 2 2 1 2 1

1 3 22

C P x C L aP xdu C Pdx k GA EI2EI

du C C x C Lxdx k GA EI2EI

+ ++= − +

= − + (p)2

3

2

a P2EI

+



( ) ( ) ( ) ( ) ( )

( ) ( )

Integrate once relations (p):

(q)

(q)

3 21 2 3 1 2 3 1 2 3 1

1 733 2 2

23 22 3 12 1 2 1 2 1

1 833 2 2 2

C P x C P x C L aP xu x C

k GA 6EI 2EI

a P xC x C x C Lxu x Ck GA 6EI 2EI 2EI

+ + += − + +

= − + + +

( ) ( ) ( ) ( )( ) ( )

2 7 8relations (q)

: Resolve constants C ,C ,C by exploiting the boundary conditions (translations):

at :

STEP 4

(r)

at

1 13 1 3 7

23 1

1. u x1 x 0 u 0 0 C 0

2. u x1 x

= = ⎯⎯⎯⎯⎯→ =

= ( ) ( )

( ) ( ) ( ) ( )

relations (q)

relations (q)

: (s)

Translational continuity condition at :

232 32 2

3 83 2 2

1 21 3 3

33 3

83 2

a LPC L C LL u L =0 C 0k GA 3EI 2EI

3. x a u a =u a

P a P aCk GA 6EI

⎯⎯⎯⎯⎯→ + + + =

=

⎯⎯⎯⎯⎯→ = −

( )

(t)

From relations (t), (s) , ... (u)2 2

3 22 33 2

3

P a 3EI3aL a 2kLC k u x11 k2L GALλ

⎛ ⎞− +⇒ =− = ⇒⎜ ⎟⎜ ⎟+⎝ ⎠



( )For a rectangular cross section b h and ν=1/3 we have 23

2 23h 1bhI , k

12 4Lν+

→ × = =

( )( )

( )( )

( )

Frοm relations (n),(u),(c),(d) we get the expressions of shear forces and bending moments:

21 3

1 2 3 33

22 3

23

12

P a 3a aQ C C P 2k P2L 1 k L L

P a 3a aQ C 2k2L 1 k L L

M C

→

⎡ ⎤⎛ ⎞= = + = − − + +⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞= = − − +⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦

= ( ) ( )( ) ( )

1 1 3 2 1 3 12

2 1 4 2 12

x C C L x P a x

M C x C C L x

+ = − − − −

= + = − −



( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )

The reactions of the beam are:

210 3

3 33

22L 3

3 3

210 3

2 2 3 32

2L2 2

P a 3a aR Q 0 2k P2L 1 k L L

P a 3a aR Q L 2k2L 1 k L L

P a 3a aM M 0 C L P a 2k P a2 1 k L L

M M L

→

⎡ ⎤⎛ ⎞= = − − + +⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤⎛ ⎞= − = − +⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦

⎡ ⎤⎛ ⎞= = − − = − + −⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦

= =

For small h/L ratios (h/L<10) which usually occur in practice, shear deformations influence negligibly internal actions and reactions of beams

0

→

02M

L x3

P3

A L x1

O

a b

03R L

3R


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NONUNIFORM TORSION, UNIFORM SHEAR AND TIMOSHENKO ...

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